Generally, the multi-parameter estimation protocol is harder than the single-parameter estimation protocol since we cannot always achieve the optimal estimation of multiple parameters simultaneously encrypted in a given quantum system. This is mainly due to the incompatibility of quantum measurements [1]}. Consequently, the quantum limits of accuracy cannot usually be achieved. To saturate the quantum Cramér-Rao bound in multiparameter estimation problems, we need to know the influence of this incompatibility on our estimation problem. This is efficiently done by using a recently introduced quantity named quantumness [2]}, [3]}. It is a scalar quantity that evaluates the asymptotic incompatibility of a simultaneous estimate of several parameters and is expressed as follows
\({\cal R}_{ \theta }:=\parallel 2i\mathbb {F}_{ \theta }^{-1}U_{ \theta }\parallel _{\infty },\) 
